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Chapter 3: Problem 19

Factor the polynomial completely, and find all its zeros. State the multiplicity of each zero. $$P(x)=x^{2}+25$$

Short Answer

Expert verified
Factor: \((x + 5i)(x - 5i)\). Zeros: \(-5i\) and \(5i\), each with multiplicity 1.

Step by step solution

01

Identify the type of polynomial

The polynomial given is \( P(x) = x^2 + 25 \). This is a quadratic polynomial lacking the linear term, therefore it resembles the form \( x^2 + c \).
02

Recognize it as a sum of squares

The polynomial \( x^2 + 25 \) can be recognized as a sum of squares: \( x^2 + 5^2 \). Recall that a sum of squares does not factor into real factors, but can be expressed using imaginary numbers.
03

Factor using complex numbers

To factor \( x^2 + 25 \) using complex numbers, we use the identity for the sum of squares: \( x^2 + a^2 = (x + ai)(x - ai) \). Here, \( a = 5 \). Therefore, the factorization is \( P(x) = (x + 5i)(x - 5i) \).
04

Determine the zeros

To find the zeros of the polynomial, set \( (x + 5i)(x - 5i) = 0 \). This results in \( x + 5i = 0 \) or \( x - 5i = 0 \). Solving these equations gives the zeros \( x = -5i \) and \( x = 5i \).
05

State the multiplicity of each zero

Each zero \( x = -5i \) and \( x = 5i \) arises from a linear factor occurring exactly once, so each zero has a multiplicity of 1.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quadratic Polynomials
Quadratic polynomials are mathematical expressions of the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants and \( a eq 0 \). Typically, they represent symmetrical curves in a graph known as parabolas. In this exercise, the polynomial \( P(x) = x^2 + 25 \) is a simple quadratic without the linear \( bx \) term. Thus, the expression is often called a "pure" or "basic" quadratic. Understanding quadratic polynomials:
  • The highest degree term is \( x^2 \), making it a second-degree polynomial.
  • Coefficients \( a \) and \( c \) dictate the shape and position of the parabola on the Cartesian plane.
  • In the case of \( x^2 + 25 \), \( a = 1 \), \( b = 0 \), and \( c = 25 \).
While most quadratics can be factored into real numbers, some like \( x^2 + 25 \) require a more imaginative approach leveraging the concept of complex numbers, since real factorization isn't possible here.
Complex Numbers
Complex numbers extend our understanding of numbers beyond the real axis. They consist of a real part and an imaginary part, typically expressed as \( a + bi \), where \( i \) is the imaginary unit defined by \( i^2 = -1 \).Key insights into complex numbers:
  • Real components are numbers we typically encounter on the number line, like 3 or -7.
  • The imaginary part involves \( i \), enabling solutions to equations where real numbers alone fall short, such as square roots of negative numbers.
  • In our example, solutions use the imaginary unit, introducing new possibilities for factoring quadratics like \( x^2 + 25 \).
Here, we used complex numbers to redefine the sum of squares into factors involving imaginary parts: \( (x + 5i)(x - 5i) \). It's a vital tool for understanding polynomial structures unseen in regular numerical solutions.
Sum of Squares
The expression \( x^2 + 25 \) is recognized as a sum of squares, which in mathematics refers to expressions of the form \( a^2 + b^2 \). Unlike the difference of squares, the sum is not factorable using real numbers alone. It requires a dive into complex numbers.How to handle the sum of squares:
  • In non-real solutions, it uses the identity: \( x^2 + a^2 = (x + ai)(x - ai) \).
  • This approach leads to complex factors if \( a \) and \( b \) are positive real numbers.
  • For \( x^2 + 25 \), simplifying it to \( x^2 + 5^2 \), the factors become \( (x + 5i)(x - 5i) \).
Utilizing the sum of squares mechanism with imaginary numbers allows us to break down the polynomial into its constituent parts that reveal the zeros, paving the way for solving equations that seem insoluble in the realm of real numbers alone.
Polynomial Zeros
Zeros of a polynomial are the values of \( x \) that make the polynomial equal to zero, effectively the solutions or roots of the polynomial equation. For the polynomial \( P(x) = x^2 + 25 \), the zeros are determined using its factorization.Understanding polynomial zeros:
  • They indicate points where the graph of the polynomial intersects the x-axis when real.
  • Zeros are found by setting each factor of the polynomial to zero and solving for \( x \).
  • In our context, factorization was made possible through the use of complex numbers, resulting in zeros: \( x = -5i \) and \( x = 5i \).
Each zero has a multiplicity, which denotes how often it appears as a solution. Here, both zeros \( -5i \) and \( 5i \) each appear once, giving them a multiplicity of 1. Recognizing polynomial zeros, even complex ones, helps us understand deeper properties of the polynomial's behavior and shape.

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